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One Actual Seed for Both Native Orders

Abstract

The complete native two-order seed uses one actual cycle and one actual inventory throughout.

Theorem 1.1 (Earn the Complete Same-Cycle Seed).

Lean statement: D5/S3/Combinatorics/Graph/PrefixReversal/NativeActualJ2Seed.native_actual_j2_seed

Proof. Machine-checked in Lean as D5/S3/Combinatorics/Graph/PrefixReversal/NativeActualJ2Seed.native_actual_j2_seed (✓ std3). ∎

Source. Repository-derived.

Commentary.

For every m at least five and every supplied two-label frame, one raw F2 Hamilton witness is selected and mapped once. Its six joint pieces and both three-piece half families retain their indexed native endpoint tables. The joint cuts are the disjoint union of the two half cut groups; all three support and filtered-edge permutations refer to that same cycle. The seed earns the protected-a edge invariant with the actual final two selected labels, the remaining-b edge invariant outside the selected set, and the selected-c invariant. The protected pair correspondence follows from the raw zero-port coordinates and the frame relabelling, without weakening the excluded pair to arbitrary selected labels. The opposite half retains its exceptional literal barred table. This is a two-label seed, distinct from the crossing-state shape required starting at the third order. The all-order construction remains a separate obligation.

References